Further results on sequetially additive graphs

Abstract

Given a graph G with p vertices, q edges and a positive integer k, a k-sequentially additive labeling of G is an assignment of distinct numbers k, k + 1, k + 2, . . . , k + p + q − 1 to the p + q elements of G so that every edge uv of G receives the sum of the numbers assigned to the vertices u and v. A graph which admits such an assignment to its elements is called a k-sequentially additive graph. In this paper, we give an upper bound for k with respect to which the given graph may possibly be k-sequentially additive using the independence number of the graph. Also, we prove a variety of results on k-sequentially additive graphs, including the number of isolated vertices to be added to a complete graph with four or more vertices to ∗Present address: School of ITMS, University of Ballarat, P.O. Box 663, Ballarat, Victoria 3353, Australia. The work reported in the paper was initiated when the first author was visiting School of Electrical Engineering and Computer Science, University of Newcastle, Australia, during October-December, 2003. Partial travel assistance was given by International Mathematical Union (IMU). 252 S.M. Hegde and M. Miller be simply sequentially additive and a construction of an infinite family of k-sequentially additive graphs from a given k-sequentially additive graph.